<p>Suppose \(n \in \mathbb{Q}\). The Binomial Theorem can be generalized to the following:</p>
<p style="text-align: center">\[(a+b)^n = a^n \left( 1 + n \left( \frac{b}{a} \right) + \frac{n(n-1)}{2!} \left( \frac{b}{a} \right)^2 + \dots \right)\]</p>
<p>Notes:</p>
<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text">This is an infinite series.</span></div></li><li><div class="list-item-content"><span class="list-item-text">This is only valid for&nbsp;\(\left| \frac{b}{a} \right| &lt; 1\).</span></div></li><li><div class="list-item-paragraph">The derivation makes use of the Maclaurin series expansion.</div></li></ol>

Suppose \(n \in \mathbb{Q}\). The Binomial Theorem can be generalized to the following:

\[(a+b)^n = a^n \left( 1 + n \left( \frac{b}{a} \right) + \frac{n(n-1)}{2!} \left( \frac{b}{a} \right)^2 + \dots \right)\]

This is only valid for \(\left| \frac{b}{a} \right| < 1\).

The derivation makes use of the Maclaurin series expansion

The derivation makes use of the Maclaurin series expansion.

Binomial Expansion & Maclaurin Series

Maclaurin Series and Power Series

International Baccalaureate Higher Level Mathematics

Binomial series as Maclaurin's series

Binomial series as Maclaurin's series